Oscillation noise-induced

Sosnovtseva, O.V., Fomin, A. I., Postnov, D.E., and Anishchenko, V.S. Clustering of noise-induced oscillations. Phys Rev E Stat Nonlin Soft Matter Phys 2001, 64 026204. 

Bulsara, A., and Gammeitoni, L Tuning in to noise. Phys. Today 1996,1996 39-45. Longtin, A., and Hinzer, K. Encoding with bursting, subthreshold oscillations, and noise in mammalian cold receptors. Neural Comput 1996,8 215-255. Mosekilde, E., Sosnovtseva, O.V., Postnov, D., Braun, H.A., and Huber, M.T. Noise-activated and noise-induced rhythms in neural systems. Nonlin Stud 2004,11 449-467. 

Samiolov M, Plyasunov S, Arkin AP. Stochastic amplification and signaling in enzymatic futile cycles through noise-induced bistability with oscillations. Proc. Natl. Acad. Sci. U.S.A. 2005 102 2310-2315. 

Comparing Eq. (110) with Eq. (66), we find that both equations have extra terms (the e or e terms) which make the solutions oscillatory, but the physical reason for oscillations is different in both cases. In Eq. (66) different from zero e comes from the nonzero initial value of the second-harmonic mode intensity, while in Eq. (110) the nonzero value of e comes from the quantum noise. We can interpret this fact in the following way. It is the spontaneous emission of photons, or vacuum fluctuations of the second harmonic mode, that contribute to the nonzero value of the initial intensity of the second harmonic mode and lead to the periodic evolution. This means that the very small quantum fluctuations can cause macroscopic effects, such as quantum-noise-induced macroscopic revivals [38], in the nonlinear process of second-harmonic generation. 

To characterize the level of coherence of noise-induced excitations we analyze the time evolution of the activator concentration x in the FitzHugh-Nagumo model, see Fig. 1.6. In this representation the excitation loops shown previously in Fig. 1.4 become spikes spaced out by intervals during which the system performs noisy relaxation oscillations aroimd its stable state. The phenomenon of coherence resonance manifests itself in the three realizations of x t) for different noise intensities given in Fig. 1.6. For very low noise intensity (upper panel) an excitation is a rare event which happens at random times. In the panel at the bottom, for high noise intensity, the systems fires more easily but still rather randomly. In the panel in the center instead, at an optimal noise intensity, the system fires almost periodically. 

The typical oscillation period for the system is given by the mean interspike time interval (ISI) tp) between two successive noise-induced excitations over many realizations, see enlargements in Fig. 1.6. To it we associate as error the standard deviation. If the system fires regularly, say for simplicity periodically, then the error associated to tp is zero and consequently the ratio of the standard deviation srd tp) to its mean value tp), i.e. the normalized fluctuations... 

Z. Hou and H. Xin. Noise-induced oscillations and stochastic resonance in an autonomous chemical reaction system. Phys. Rev. E, 60 6329, 1999. 

M. Zaks, X. Sailer, L. Schimansky-Geier, and A. Neiman. Noise induced complexity From subthreshold oscillations to spiking in coupled excitable systems. CHAOS, 15 026117, 2005. 

The influence of noise on a dynamical system may have two counteracting effects. On the one hand if the underlying deterministic systems is already oscillatory, like a limit cycle oscillator or a chaotic oscillator, one expects these oscillations to become less regular due to the influence of the noise. On the other hand oscillatory behavior can also be generated by the noise in systems which deterministically do not show any oscillations. A prominent example are excitable systems but also the noise induced hopping between the attractors in a bistable system can be considered as oscillations [1]. 

Time-delayed feedback control has also been applied to purely noise-induced oscillations in a regime where the deterministic system rests in a steady state. It has been shown that in this way both the coherence and the mean frequency of the oscillations can be controlled in simple models [7-9, 49, 50] as well as in spatially extended systems [51-53]. 

We choose the control parameters U and a such that the deterministic system exhibits no oscillations but is very close to a bifurcation thus yielding it very sensitive to noise. The transition from stationarity to oscillations in the system may occur either via a Hopf or via a saddle-node bifurcation on a limit cycle as depicted in the bifurcation diagram of Fig. 5.9. The different nature of these two bifurcations is reflected in the effect noise has in each case. The local character of the Hopf bifurcation is responsible for noise-induced high frequency oscillations of strongly varying amplitude around the stable fixed point. We try to characterize basic features of these oscillations such as coherence and time scales. The need to be able to adjust these features as one wishes will lead to the application of the time-delayed... 

To conclude, noise-induced front motion and oscillations have been observed in a spatially extended system. The former are induced in the vicinity of a global saddle-node bifurcation on a limit cycle where noise uncovers a mechanism of excitability responsible also for coherence resonance. In another dynamical regime, namely below a Hopf bifurcation, noise induces oscillations of decreasing regularity but with almost constant basic time scales. Applying time-delayed feedback enhances the regularity of those oscillations and allows to manipulate the time scales of the system by varying the time delay t. 

In summary, noise induces oscillations in the system, which would otherwise rest in its inhomogeneous fixed point. With growing noise intensity the... 

In order to control the noised-induced patterns, we will now use the method of time-delayed feedback which was previously applied successfully in deterministic chaos control of this particular system [47] as well as for control of noise-induced oscillations in simple models [7-9] without spatial degrees of freedom. 

To get a first impression whether or not this control force is able to change the temporal regularity of the noise-induced oscillations we fix = 0.1, Da = 10 , as in the upper panel of Fig. 5.21, and calculate the... 

A. G. Balanov, N. B. Janson, and E. Scholl Control of noise-induced oscillations by delayed feedback, Physica D 199, 1 (2004). 

J. Pomplun, A. Amann, and E. Scholl Mean field approximation of time-delayed feedback control of noise-induced oscillations in the Van der Pol system, Europhys. Lett. 71, 366 (2005). 

J. Hizanidis, A. G. Balanov, A. Amann, and E. Scholl Noise-induced oscillations and their control in semiconductor superlattices, Int. J. Bifur. Chans 16, 1701 (2006). 

Noise-induces oscillations and excitable systems. These systems represent a separate class exhibiting synchronization. Without noise, these systems are stable and do not oscillate. In the presence of noise the dynamics is similar to the dynamics of noisy self-... 

Systems with delays. Oscillators with internal delays [18], delays in coupling [53], and delayed-feedback control of chaotic and noise-induced oscillations [6, 45] represent another field of actual research. 

The frequency noise power spectral density of a SL typically exhibits a 1/f dependence below 100 kHz and is flat from 1 MHz to well above 100 MHz. Relaxation oscillations will induce a pronounced peak in the spectrum above 1 GHz. The "white" spectral component represents the phase fluctuations that are responsible for the Lorentzian linewidth and its intensity is equal to IT times the Lorentzian FWHM.20 xhe 1/f component represents a random walk of the center frequency of the field. This phase noise is responsible for a slight Gaussian rounding at the peak of the laser field spectrum and results in a power independent component in the linewidth. Figure 3 shows typical frequency noise spectra for a TJS laser at two power levels. 

Noise-induced transitions have been studied theoretically in quite a few physical and chemical systems, namely the optical bistability [12,13,5], the Freedricksz transition in nematics [14,15,16,5], the superfluid turbulence in helium II [17], the dye laser [18,19], in photochemical reactions [20], the van der Pol-Duffing oscillator [21] and other nonlinear oscillators [22]. Here I will present a very simple model which exhibits a noise-induced critical point. The so-called genetic model was first discussed in [4]. I will not describe its application to population genetics in this paper, see [5] for this aspect, but use a chemical model reaction scheme ... 

We shall more particularly be interested in the effect of the speed of the external noise on the noise-induced Hopf bifurcation. We consider successively the two cases where the correlation time of the noise is shorter than the period of oscillation T or longer. 

The observed fluctuations are very sensitive to stirring. The pronounced decrease of the oscillation period with stirring rate is accompanied by a marked increase of the fluctuation amplitude. Figure 2 illustrates our interpretation in terms of the contraction of the limit cycle from its deterministic limit 1-2-3-4 to the stochastic limit cycle l -2 -3 -4. The latter is dominated by noise-induced transitions. The interaction between local (stochastic) and global dynamics is seen to be profound. 

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